Some Estimates of Fourier Transforms

This work consists of two independent parts. In the first part we prove several results related to the theorem of Logvinenko and Sereda on determining sets for functions with Fourier transforms supported in a parallelepiped. We obtain a polynomial instead of exponential bound in this theorem, and we extend it to the case of functions with Fourier transforms supported in the union of a bounded number of parallelepipeds. When dimension d = 1 we also consider the case of infinitely many lacunary intervals. We generalize the Zygmund theorem for lacunary series whose Fourier coefficients are replaced with polynomials of uniformly bounded degree. We give also a necessary condition for the support of Fourier transforms for which the Logvinenko-Sereda theorem still holds. In the second part we prove that the L²([0,1]d x SO(d)) norm of periodizations of a function from L¹(ℝd) is equivalent to the L²(ℝd) norm of the function itself in higher dimensions. We generalize the statement for functions from Lp(ℝd) where 1 ≤ p < (2d)/(d + 2) spirit of the Stein-Tomas theorem. We also show that the following theorem due to M. Kolountzakis and T. Wolff does not hold if dimension d = 2: if periodizations of a function from L¹(ℝd) are constants, then the function is continuous and bounded provided that the dimension d is at least three.